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Abstract

Dynamical instrument limitations, such as finite detection bandwidth, do not simply add statistical errors to fluctuation measurements, but can create significant systematic biases that affect the measurement of steady-state properties. Such effects must be considered when calibrating ultra-sensitive force probes by analyzing the observed Brownian fluctuations. In this article, we present a novel method for extracting the true spring constant and diffusion coefficient of a harmonically confined Brownian particle that extends the standard equipartition and power spectrum techniques to account for video-image motion blur. These results are confirmed both numerically with a Brownian dynamics simulation, and experimentally with laser optical tweezers.

Figures (6)

(a) Brownian dynamics simulation results for measured variance as a function of exposure time. Data has been rescaled and plotted alongside S(α), the motion blur correction function of Eq. (7), showing excellent agreement within the expected error. The step size of the simulation is by 1µs, which is less than 0.01τ for all three simulations. The different simulation settings are: (i) 1.6µm bead radius, k=0.05 pN/nm, τ=0.537ms, (ii) 0.4µm bead radius, k=0.05 pN/nm, τ=0.134 ms, (iii) (1.6µm bead radius, k=0.0125 pN/nm, τ=2.148 ms) (b) Histogram of measured positions for simulation run (c) for an exposure time of 4 ms. It is a Gaussian distribution as expected [16]. The normal curve with the predicted variance is superimposed showing excellent agreement. The expected distribution for an ideal “blur-free” measurement system is superimposed as a dotted line.

Spring constant vs. power for a single bead in the optical trap. The naïve equipartition measured spring constant with 1 ms and 2 ms exposure times (red triangles and blue squares, respectively) is compared with the blur corrected spring constant (black circles). The dashed blue and red lines going through the uncorrected data represent non-linear fits to the blur model assuming a linear relationship between k and laser power, i.e. k=cP, as discussed in subsection (5.3). The values obtained from these fits for c and γ agree within error with the “black circle” values obtained by varying the exposure time.

Experimentally measured variance as a function of the high pass filter cutoff frequency shows a linear relation (line), which can be extrapolated to 0 Hz to reliably estimate the drift-free variance. The variance without filtering (cross) is 100 nm2, while the extrapolated variance (star) is 78.5nm2

A log-log plot of the one-sided power spectrum (dots) for a trapped bead, with theoretical models produced from a least squares fit to the data (blur-corrected and aliased, Eq. (23) solid line; naïve, Eq. (13) blue dashed line; naïve aliased, green dotted line). The effect of the motion blur correction function S(α) is readily apparent from the clear discrepancy between the solid red and dotted green lines.

Fractional deviation of the power spectrum data obtained by dividing the experimentally measured values (dots in Fig. 5) by the fit obtained with the blur-corrected and aliased model (solid red line in Fig. 5). Left: Scatter plot demonstrating the quality of the fit; the two dashed red lines indicate the estimated standard deviation from unity of
1128
[14]. Right: Histogram of the fractional deviation data overlaid with a Gaussian distribution with a standard deviation of
1128
(solid red line).